### Can a small forcing create Kurepa trees?

by Jin and Shelah. [JiSh:563]

Annals Pure and Applied Logic, 1997

In the paper we probe the possibilities of creating a Kurepa
tree in a generic extension of a model of CH plus no Kurepa trees by
an omega_1-preserving forcing notion of size at most
omega_1 . In the first section we show that in the Levy model
obtained by collapsing all cardinals between omega_1 and a
strongly inaccessible cardinal by forcing with a countable support
Levy collapsing order many omega_1-preserving forcing notions of
size at most omega_1 including all omega-proper forcing
notions and some proper but not omega-proper forcing notions of
size at most omega_1 do not create Kurepa trees. In the second
section we construct a model of CH plus no Kurepa trees, in which
there is an omega-distributive Aronszajn tree such that forcing
with that Aronszajn tree does create a Kurepa tree in the generic
extension. At the end of the paper we ask three questions.

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